JournalsjncgVol. 10, No. 4pp. 1269–1301

Symmetric monoidal noncommutative spectra, strongly self-absorbing CC^*-algebras, and bivariant homology

  • Snigdhayan Mahanta

    University of Regensburg, Germany
Symmetric monoidal noncommutative spectra, strongly self-absorbing $C^*$-algebras, and bivariant homology cover
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Abstract

Continuing our project on noncommutative (stable) homotopy we construct symmetric monoidal \infty-categorical models for separable CC^*-algebras SC\mathtt{SC_\infty^*} and noncommutative spectra NSp\mathtt{NSp} using the framework of Higher Algebra due to Lurie. We study smashing (co)localizations of SC\mathtt{SC_\infty^*} and NSp\mathtt{NSp} with respect to strongly self-absorbing CC^*-algebras. We analyse the homotopy categories of the localizations of SC\mathtt{SC_\infty^*} and give universal characterizations thereof. We construct a stable \infty-categorical model for bivariant connective E\mathtt E-theory and compute the connective E\mathtt E-theory groups of O\mathcal{O}_\infty-stable CC^*-algebras. We also introduce and study the nonconnective version of Quillen's nonunital K\mathtt K'-theory in the framework of stable \infty-categories. This is done in order to promote our earlier result relating topological T\mathbb T-duality to noncommutative motives to the \infty-categorical setup. Finally, we carry out some computations in the case of stable and O\mathcal{O}_\infty-stable CC^*-algebras.

Cite this article

Snigdhayan Mahanta, Symmetric monoidal noncommutative spectra, strongly self-absorbing CC^*-algebras, and bivariant homology. J. Noncommut. Geom. 10 (2016), no. 4, pp. 1269–1301

DOI 10.4171/JNCG/260